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Difference between revisions of "2008 UNCO Math Contest II Problems/Problem 4"

(Created page with "== Problem == 5. The sum of <math>400, 3, 500, 800</math> and <math>305</math> is <math>2008</math> and the product of these five numbers is <math>146400000000 = 1464 \times 10...")
 
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== Problem ==
 
== Problem ==
  
 +
In the figure there are <math>8</math> line segments drawn from
 +
vertex <math>A</math> to the base <math>BC</math> (not counting the segments <math>AB</math> or <math>AC</math>).
  
−
5. The sum of <math>400, 3, 500, 800</math> and <math>305</math> is <math>2008</math> and the product of these five numbers is
+
<asy>
−
<math>146400000000 = 1464 \times 10^8.</math>
+
for (int x=0;x<11;++x){
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draw((5,15)--(x,0),dot);
 +
}
 +
draw((0,0)--(10,0),black);
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draw((10/6,5)--(10-10/6,5),black);
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draw((20/6,10)--(10-20/6,10),black);
 +
MP("A",(5,15),N);MP("B",(0,0),W);MP("C",(10,0),E);
 +
</asy>
  
−
(a) Determine the largest number which is the product of positive integers whose sum is <math>2008</math>.
+
(a) Determine the total number of triangles of all sizes.
  
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(b) Determine the largest number which is the product of positive integers whose sum is <math>n</math>.
+
(b) How many triangles are there if there are <math>n</math> lines
 +
drawn from <math>A</math> to <math>n</math> interior points on <math>BC</math>?
  
  
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{{UNCO Math Contest box|n=II|year=2008|num-b=3|num-a=5}}
 
{{UNCO Math Contest box|n=II|year=2008|num-b=3|num-a=5}}
  
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[[Category:Intermediate Algebra Problems]]
+
[[Category:Intermediate Geometry Problems]]

Revision as of 21:55, 19 October 2014

Problem

In the figure there are $8$ line segments drawn from vertex $A$ to the base $BC$ (not counting the segments $AB$ or $AC$).

[asy] for (int x=0;x<11;++x){ draw((5,15)--(x,0),dot); } draw((0,0)--(10,0),black); draw((10/6,5)--(10-10/6,5),black); draw((20/6,10)--(10-20/6,10),black); MP("A",(5,15),N);MP("B",(0,0),W);MP("C",(10,0),E); [/asy]

(a) Determine the total number of triangles of all sizes.

(b) How many triangles are there if there are $n$ lines drawn from $A$ to $n$ interior points on $BC$?


Solution

See Also

2008 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
All UNCO Math Contest Problems and Solutions
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